from a Mechanical Point of View. 701 



equations are * 



Da _ Y dp ( ,d0 (dhi d?\ 



' Ttf 



T)a ^ dp ,d6 Id? a 



Dv v dp , ,ae id-v d'-v 



"Bt^^-J^^dij + ^W+W 



, „ du dv 



where = ^- + - r . 



ctx dy 



If the pressure is independent of density and if the inertia 

 terms are neglected, these equations are satisfied provided 



P X + // tf 0/^ = 0, pY+fi' dOjdy = 0. 



In the case of real viscous fluids, there is reason to think 

 that fx =^fju. Impressed forces are then required so long as 

 the fluid is moving. The supposition that p is constant being 

 already a large departure from the case of nature, we may 

 perhaps as well suppose /^' = 0, and then no impressed bodily 

 forces are called for either at rest or in motion. 



If we suppose that the motion in (7) is steady in the hydro- 

 dynamical sense, u + iv must be independent of t, so that the 

 elimination of tj-\-irj between (5) and (6) must carry with it 

 the elimination of t. This requires that df/dt in (6) be a 

 function of /, and not otherwise of t and g + irj ; and it 

 follows that (5) must be of the form 



.!• + «/ = F 1 {< + F 2 (£+;,)}, .... (9) 



where F l5 F 2 denote arbitrary functions. Another form 

 of (9) is 



F 3 (x + iy) = t + ¥ 2 (£+i V ) (10) 



For an individual particle F 2 {^+ir]) is constant, say a + ib. 

 The equation of the stream-line followed by this particle 

 is obtained by equating to ib the imaginary part of 



F 3 + ?». 



As an example of (9), suppose that 



x + iy = c sin {it + !; + iy] , .... (11) 

 so that 



x ~ c sin £ cosh (rj + 1) , y = c cos £ sinh (rj + 1), . (12) 



whence on elimination of t we obtain (1) as the equation of 

 the stream-lines. 



* Stokes. Camb. Trans. 1850 ; Mathematical and Physical Papers, 

 ■vol. iv. p. 11. It does not seem to be generally known that the laws of 

 dynamical similarity for viscous fluids were formulated in this memoir. 

 Reynolds's important application was SO years later. 



